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X. Chen, E. Daus, A. Jüngel:
"Global existence analysis of cross-diffusion population systems for multiple species";
in: "ASC Report 18/2016", herausgegeben von: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, S. 1 - 30.

The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homo-geneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed bal-ance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager´s principle in thermodynamics. Under detailed balance (and without reaction), the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold.

Population dynamics, Shigesada-Kawasaki-Teramoto system, competition model, detailed balance, entropy method, global existence of weak solutions, Onsager´s principle.

http://www.asc.tuwien.ac.at/preprint/2016/asc18x2016.pdf